Numerical computation of the Rosenblatt distribution and applications

TL;DR

This paper introduces new methods for accurately computing the Rosenblatt distribution, including characteristic functions, eigenvalues, and density, with applications demonstrated through simulations of Gaussian processes with long-range dependence.

Contribution

It provides novel expressions for the characteristic function, an efficient eigenvalue approximation, and a computational algorithm for the Rosenblatt distribution.

Findings

New characteristic function expressions derived

Eigenvalue approximation method developed

Monte-Carlo simulations confirm distribution appearance

Abstract

The Rosenblatt distribution plays a key role in the limit theorems for non-linear functionals of stationary Gaussian processes with long-range dependence. We derive new expressions for the characteristic function of the Rosenblatt distribution. Also we present a novel accurate approximation of all eigenvalues of the Riesz integral operator associated with the correlation function of the Gaussian process and propose an efficient algorithm for computation of the density of the Rosenblatt distribution. We perform Monte-Carlo simulation for small sample sizes to demonstrate the appearance of the Rosenblatt distribution for several functionals of stationary Gaussian processes with long-range dependence.